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PoS(CORFU2019)099

Ignatios Antoniadis1,2, Yifan Chen3, George K. Leontaris∗ 4

1LPTHE Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France E-mail:antoniad@lpthe.jussieu.fr

2ITP, Albert Einstein Center, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland 3CAS Key Laboratory of Theoretical Physics, Insitute of Theoretical Physics, Chinese Academy

of Sciences, Beijing 100190, P.R.China E-mail:yifan.chen@itp.ac.cn

4Theory Section, Physics Department, University of Ioannina, GR-45110 Ioannina, Greece

E-mail:leonta@uoi.gr

The problem of moduli stabilisation is discussed in type IIB/F-theory. Considering a configu-ration of three intersecting D7 branes with fluxes, it is shown that higher loop effects induce logarithmic corrections to the Kähler potential which can stabilise Kähler moduli, in particular the internal volume of the compactification space in 4 dimensions. It is also explained how D-term contributions to the effective potential stabilise the rest of the Kähler moduli and uplift to de Sitter vacua.

Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2019)

31 August - 25 September 2019 Corfu, Greece

Speaker.

c

Copyright owned by the author(s) under the terms of the Creative Commons

Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/

https://doi.org/10.7892/boris.147899

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PoS(CORFU2019)099

1. Introduction

Nowadays there is adversarial debate amongst high energy physics community whether the

string theory landscape contains de Sitter (dS) vacua. No go theorems such as [1], and

swamp-land conjectures [2] cast doubts on their existence in consistent quantum theories1 and disfavour

cosmological inflation in effective field theory models of string origin. Instead, there are

sugges-tions [4] that quintessence models where the cosmological constant varies over time satisfy current

observational constraints. If so, the present acceleration phase shall come to an end and eventually the rapid expansion of the universe will terminate in the distant future. However, string quantum corrections which are anticipated to play essential role in shaping the final form of the effective theory and the scalar potential, have not been fully explored yet. Therefore, before abandoning the attractive de Sitter solutions which abide with successful inflationary models, it is worth ex-ploring further the ‘landscape’ of string compactifications and the implications of various localised defects (such as D-branes) on the emerging effective theories. This presentation will focus on the investigation of dS vacua in type IIB superstring theory in the presence of D-branes.

There are several ways of building de Sitter vacua in the literature. Most common are those

based on non-perturbative corrections, such as the KKLT mechanism [5] and the so-called large

volume scenario [6]. These constructions are much debadable however, due to the fact that they rely

on non-perturbative effects which cannot be controlled at the full string level (some improvements

of the original models have appeared using the nilpotent chiral multiplets [7], leading to a new

mechanism of uplifting the vacua in the stringy landscape [8]).

In this work, de Sitter vacua will be sought only from sources inducing perturbative correc-tions. It will be argued that higher derivative terms in the ten-dimensional string effective action generate upon compactification a four-dimensional Einstein-Hilbert term localised in the internal space, which in turn induces logarithmic corrections to the scalar potential via loop effects. At this point, it is worth emphasising that such corrections are standard in the presence of D7-branes; they appear as quantum corrections either in the world-sheet or in the string coupling perturbation

theory and have also been studied in different contexts in the past [9, 10]. Such contributions,

break the no-scale structure of the Kähler potential and lead to an effective theory with some

Käh-ler moduli stabilised, such as the internal six-dimensional volume [11]. On the other hand, D-term

contributions associated with abelian symmetries which are present for instance in configurations of intersecting D7-branes, work as an “uplifting” mechanism, stabilising the remaining Kähler

moduli and ensuring the existence of de Sitter vacua [11].

2. Prelude

The aim of this work is to propose a solution to the moduli stabilisation problem in II-B string theory and to examine whether a dS minimum exists in consistent theories (i.e. those having an ultraviolet (UV) completion), based only on perturbative corrections. To set the notation and focus

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only on what is needed from type II-B framework, the moduli fields to be used in the subsequent

analysis are briefly presented. 2

From the NS-NS sector, the relevant ingredients are the dilaton and KB-field:

gµ ν, φ , Bµ ν→ B2

From the R-R sector, we make use of the various p-form potentials: C0, Cµ ν, Cκ λ µ ν→ Cp, ; p= 0, 2, 4 .

The C0potential and the dilaton field, are used in the usual axion-dilaton combination denoted here

with S:

S= C0+ i e−φ→ C0+

i

gs,

where gsis the string coupling. In addition, we will consider a six-dimensional compactification on

a CY threefold and denote the complex structure (CS) moduli with za, and the Kähler class moduli with Ti:

CS : za, a = 1, 2, 3 . . . h(2,1), ; Kähler : Ti, i = 1, 2, 3, . . . h(1,1).

Note also that CS moduli are (2,1)-forms whilst Kähler moduli are (1,1)-forms, with their respective

number h(2,1) and h(1,1). Obviously, if moduli remain massless at low energies, they give rise to

fifth forces and create other cosmological problems in the effective field theory. Hence, the task is to generate a potential and assure positive mass-squared for all moduli fields.

The subsequent analysis will be presented in the framework of type II-B N = 1 effective super-gravity. In this context, the two basic ingredients related to the present work are the superpotential of the moduli fields and the Kähler potential. Type II-B compactifications on six-dimensional CY manifolds give rise to N = 2 supersymmetry in four dimensions. This is broken to N = 1 in the

presence of 3-form fluxes and orientifold planes satisfying a set of conditions (see e.g. [12]). To

construct the superpotential we need the field strengths derived from the KB field and the p-form potentials. We also need to introduce the holomorphic (3,0)-form Ω which is a function of the CS moduli

Fp:= d Cp−1, H3:= d B2 ⇒ G3:= F3− S H3, Ω = Ω(za) .

Then, the fluxed induced superpotential W0, at the classical level, is given by the well-known

formula [13]:

W0=

Z

G3∧ Ω(za) . (2.1)

Clearly, the perturbative superpotential W0 is a holomorphic function and depends on the

axion-dilaton modulus S, and the CS moduli za. Thus, we apply the supersymmetric conditions, w.r.t. these fields za, S by setting the Kähler covariant derivatives to zero:

DzaW0= 0 , DSW0= 0 . (2.2)

2The usual abbreviations for Calabi-Yau (CY), Neveu-Schwarz (NS), Ramond (R), and Kalb-Ramobd (KB) are also used.

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The above conditions fix the moduli za, S which are present in the superpotentialW0. On

the contrary, the Kähler moduli, being (1,1)-forms, do not enter in the superpotential and thus, they remain completely undetermined at this stage. Nevertheless, we may appeal to the second ingredient of the effective supergravity which is the Kähler potential that depends logarithmically on the various moduli fields, including the Kähler moduli:

K0 = −2 ln (iV ) − ln(−i Z Ω ∧ ¯Ω) = − 3

i=1

ln(−i(Ti− ¯Ti)) − ln(−i(S − ¯S)) − ln(−i

Z

Ω ∧ ¯Ω) , (2.3)

whereV is the volume of the CY depending on the Kähler class moduli. We then specialised for

simplicity to the case of the three geometric moduli of the six dimensional torus T6corresponding

the areas of three orthogonal planes T2× T2× T2. One can now compute the effective potential

using the standard formula

Veff = eK(

I,J

DIW0KI ¯JD

¯

JW0− 3|W0|2) · (2.4)

This can be split it in two parts:

Veff = eK

{I,J}={za,S} DIW0K−1 I ¯J DJ¯W0 (DIW0= 0 , supersymmetry) (2.5) + eK

i, j Ki j 0 DiW0DjW0− 3|W0| 2 ! (= 0, no scale structure) (2.6) = 0 (2.7)

The first part of the scalar potential (2.5) involves the moduli fields za, S and it vanishes due to the

supersymmetric conditions (2.2). The second part (2.6) involves only the Kähler moduli Ti and is

also identically zero because of the no-scale structure of the Kähler potential (2.3). Hence, it is

impossible to fix the Kähler moduli at this (classical) level.

From the above discussion, it is inferred that in order to stabilise the Kähler moduli it is necessary to go beyond the classical level and include higher order effects. In fact, when quantum corrections are included they break the no-scale structure of the Kahler potential. To this end, let us first assume a generic test function depending only on a single modulus τ, representing such corrections in the form = γ f (τ), with γ a small parameter measuring their strength. This breaks

the no-scale structure and the Kähler potential takes the form in the large volume expansion3

K = −2logτ

3

2+ γ f (τ)



, V = τ32, |γ| < 1 (2.8)

Expanding in γ, the F-term potential becomes

VF∝ γ

3 f (τ) − 4τ f0(τ) + 4τ2f00(τ)

τ92

. (2.9)

As expected, when γ → 0, the no-scale structure of the Kähler potential is restored and VF≡ 0.

3

τ is assumed to be a 4-cycle modulus and thus, the internal volume has the simple formV = τ

3 2.

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PoS(CORFU2019)099

Next, let us consider the typical case of perturbation theory at large volume with a power-law

in the modulus τ, i.e., f (τ) ∝ τn. In this case, the resulting scalar potential exhibits a monotonic

behaviour in τ, VF ∝ τn−

9

2, hence there is no minimum. On the other hand, if the function f (τ) is

a logarithm of the modulus τ, the potential contains two different types of terms competing each other and in principle a minimum can exist. Indeed:

VF ∝ γ τn− 9 2  log(τ) −8 3  + · · · ⇒ ∃ (VF)minif γ < 0 , (2.10)

i.e., at a certain value of τ, the potential has a minimum provided γ < 0. Note though that this is an anti-de Sitter (AdS) vacuum and, as we will see below, additional uplifting terms will be required to obtain the desired dS minimum. Furthermore, assuming an exrta constant correction ξ in the Kähler potential: K = −2logτ 3 2 + ξ + γln(τ)  , (2.11)

it is is found that the size of the volume at the minimum is controlled by the constant parameter

µ ≡ eξ /γ:

Vmin=e

13/3

µ ; µ = e

ξ /γ.

The constant µ can be chosen so that we can have an exponentially large volume.

From the above simple example, it is obvious that the existence of a minimum depends cru-cially on the properties of the function f (τ) associated with the quantum corrections. Hence, the crux of the matter is whether a suitable type of quantum corrections exist in string theory.

3. “Localised” Einstein-Hilbert term

In this part of the presentation, the role of perturbative string loop corrections will be exam-ined. Firstly, we recall the well known world-sheet corrections, perturbative in the string slope

α0, which are found to be proportional to the Euler characteristic χ of the compactification

mani-fold. Secondly, subsequent string loop contributions will be computed which are found to be very important in the presence of D7-branes.

Starting with α0corrections, it was shown that in the large volume limit they imply a

redefini-tion of the dilaton field: [12]

e−2φ4 = e−2φ10(V + ξ) = e−12φ10( ˆV + ˆξ), (3.1)

where φDis the D-dimensional dilaton (φ10is fixed from the supersymmetry condition (2.2)) and

the last expression holds in the Einstein frame. The volume can be expressed in terms of the

imaginary parts tkof the Kähler deformations Tkas follows:

V = 1

3!κi jkt

i

tjtk, tk= −Im(Tk) = ˆtke12φ10.

Therefore, the corrections to the Kähler potential (2.3) can be accounted by a shift of the volume

by a constant ξ which is determined in terms of the Euler characteristic

ξ = − ζ (3)

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PoS(CORFU2019)099

The above corrections are associated with higher derivatives induced in the 10-dimensional effective action. Indeed, as is well known, multigraviton scattering in string theory generates higher derivative couplings. The leading terms in the effective action are those with eight derivatives and

are proportional to R4, (i.e., the Riemann tensor to the power four with appropriate contractions

of indices). Upon compactification to four dimensions, they induce an Einstein-Hilbert (EH) term which is multiplied by the Euler characteristic of the manifold.

Indeed, let us just cast a glance back to the previous works incorporating the R4 term in the

Type II-B 10-dimensional effective action. According to [14,15,16,12] the latter is given by

S ⊃ 1 (2π)7α04 Z M10 e−2φ10R(10) 6 (2π)7α0 Z M10 (−2ζ (3)e−2φ10− 4ζ (2))R4∧ e2, (3.3)

whereR(D) is the D-dimensional Ricci scalar. Compactifying six dimensions on a CY manifold

X6(so that M10= M4×X6), and taking into account the tree-level and the one-loop generated EH

terms, the ten-dimensional action reduces to

Sgrav = 1 (2π)7α04 Z M4×X6 e−2φ10R(10)+ χ (2π)4α0 Z M4 2ζ (3)e−2φ10+ 4ζ (2)R(4), (3.4)

with the Euler characteristic defined as

χ = 3

4π3

Z

X6

R∧ R ∧ R ·

Therefore, after compactification, a localised EH term is generated with a coefficient proportional to the Euler characteristic χ. From the above reduction and the form of χ, it is also deduced that this effective EH-term is possible only in four dimensions.

More precisely, the above localisation is realised in the non-compact limit (large volume) at

‘points’ in the internal space where the Euler number χ is concentrated [16]. For instance, in the

orbifold case, these are the fixed points of the orbifold group. At these points, there are localised vertices associated with the induced EH term, emitting gravitons and Kaluza-Klein (KK) excita-tions in the 6-dimensional space. A useful notion related to this mechanism is the localisation width which can be estimated by computing the graviton scattering involving two massless gravitons and

one KK excitation (see Fig.1). The computation can be done explicitly in the orbifold case, where

the term proportional to ζ (3) in the action (3.4) vanishes and the induced EH-term arises at one

loop order [16]. The result for the T6/ZNorbifold is expressed as follows:

hV2 (0,0)V(−1,−1)i = −CR 1 N2

f,k eiγkq·xf Z F d2τ τ22 Z

i=1,2,3 d2zi τ2 (h,g)

0 eα0q2F(h,g)(τ,zi),

where the subscripts in the vertices denote the ghost-picture, q is the internal (KK) momentum, CR

is a constant related to the tensor structure, whilst xf are the fixed points of the orbifold and γk is

the representation of the action of the orbifold group. The pairs (h, g) label the orbifold sectors and the prime in the sum excludes the untwisted sector (0, 0). The integration over the modulus of the

world-sheet torus τ = τ1+ iτ2is restricted as usual in the fundamental domainF of the modular

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PoS(CORFU2019)099

k

k

k

1 2 3

Figure 1:Non-zero contribution from 1-loop; 3-graviton scattering amplitude of 2 massless and 1 KK.

Referring to the relevant works for the details [11,16], one finds that the width w associated

with the localised graviton vertices is given by

w2≈ α0F(h,g)(τ, za) |min∼

ls2

N (3.5)

with ls=√α0the fundamental string length. The localisation property can be expressed in terms

of the width (3.5), by computing the amplitude in the position space ‘y’, by a Fourier transform

from the momentum space ‘q’. Indeed, it is found that the coefficient of the induced EH action

R(4)depending on the 6-dimensional internal space (in the position or momentum representation)

exhibits a Gaussian profile

CRN

w6e

−w2q2

⊥/2 (3.6)

controlled by the ‘effective’ width defined above. Here q⊥ is the internal momentum along the

directions allowed by momentum conservation. For instance, if the graviton emission ends on

7-branes, q⊥ is along the directions tranverse to their world-volume (see below). Then, the one loop

correction takes the form

4ζ (2) (2π)7α0χ Z M4×X6 e−y2/(2w2) w6 R(4), (3.7)

where N has been interpreted as the Euler characteristic χ ∼ N.

In type II-B (as well as in F-theory) context, the geometric configuration may also include D7 branes and orientifold O7 planes. In effect, there can be an exchange of KK-modes between the localised gravity position (for instance an orbifold fixed-point) and the distinct 7-branes. Since D7 branes occupy four internal dimensions, KK-modes transmitted towards each one of them

propa-gate in a 2-dimensional bulk transverse to D7 (see Fig.2). In this way, logarithmic contributions are

induced depending on the size of the 2-dimensional space transverse to D7. From the form of the

effective EH term given in (3.4) it can be seen that this correction is multiplied by the Euler

char-acteristic. Referring to the relevant works for the details [11,16], the corresponding contribution

takes the form

AS = −CRg2sT 2π sin2πN ( −γ 2+ ln R⊥ √ 2 w ! +O w 2 R2 ) , (3.8)

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PoS(CORFU2019)099

whete T is the D7 brane tension and R⊥measures the size of its transverse 2-dimensional space. In

the above formula we are interested in the R⊥dependence, which exhibits a logarithmic behaviour.

Summing up the two contributions (3.6) and (3.8), and replacing χ ∼ N as above, one finds:

4ζ (2) (2π)3χ Z M4 1 −

k e2φTkln(Rk⊥/w) ! R(4). (3.9) k k z z z k k k k 1,0 1,0 1 2,0 2,0 2 3,n 3,n 3 X X X X D7 > y=0 y=ya A worldsheet

Figure 2:Non-zero contribution from 1-loop; 3-graviton scattering amplitude of 2 massless and 1 KK propagating in 2-dimensions towards a D7 brane.

4. The effective potential

The above corrections induce new terms to the Kähler potential. It can be shown [17] that

stabilisation of the Kähler moduli requires at least three sets of D7 branes, with mutual orthogonal transverse spaces. Hence, the contributions involving the Kähler moduli are written in the form:

δ = 3

k=1

γkln(τk) .

For simplicity, we take γk= γ/2. Then, the Kähler potential takes the form

K = −2ln(√τ1τ2τ3+ ξ + γ ln (τ1τ2τ3)) ≡ −2 ln (V + ξ + γ lnV ) (4.1)

It can be easily checked that these corrections do break the no-scale structure. The F-term scalar

potential is computed using (4.1). Expanding in γ, it takes the form

VF ≈ 3γw20ln µV − 4

V3 +O(γ

3) ,

where ξ has been substituted with γ ln µ, and w0= hW0i.

In the presence of D7 branes, it is also possible that D-terms appear. These give rise to a contribution of form VD = 3

a=1 da τa  ∂K ∂ τa 2 ≈ 3

a=1 da τa3 + · · ·

with danumerical constants. Thus, the effective potential is the sum

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PoS(CORFU2019)099

Minimisation with respect to the volume implies the condition [17]:

γ w20(13 − 3 lnV ) = 2dV ; d = (d1d2d2)

1 3

The solution of the above equation determines the volume at the extrema and depends on the

double-valued Lambert W-function[17], which takes real double values in the region between −1/e

and 0. Hence, in this region the potential has a maximum and a minimum. However, not all values of this region are acceptable since the requirement for de Sitter vacua puts additional constraints. Putting all these together, the following constraints apply to the various constants

−0.007242 < ρ < −0.006738, ρ = d

γ w20µ (4.3)

In Fig. 3 the effective potential Veff is plotted versus the volume V for fixed values of the

parameter ρ. The values of ρ in the upper two curves lies within the acceptable region (4.3) and

hence they yield de Sitter minima. On the contrary, the lower curve is for a value of ρ outside the above region and thus, it corresponds to an AdS vacuum. At large volume, the potential vanishes asymptotically after passing from a maximum.

ϱ=-0.00695 ϱ=-0.00678 ϱ=-0.00669 > ϱmax 15 000 20 000 25 000  5 10 15

Figure 3:Plot of the scalar potential as a function of the volume for different values of the parameter ρ

5. Conclusions

In this presentation it has been demonstrated that perturbative corrections in a specific D7-brane configuration of type II-B string compactifications have the appropriate ingredients to

sta-bilise the Kähler moduli. The origin of these corrections come from the reduction of R4couplings

of the ten-dimensional string effective action and can exist only in 4-dimensions [18]. One may

conclude that this is an indispensable element for the existence of 4-dimensional de Sitter vacua. Acknowledgements

Work was supported in part by the Labex “Institut Lagrange de Paris”, in part by the Swiss National Science Foundation, in part by a CNRS PICS grant, and in part by the Erasmus exchange program.

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PoS(CORFU2019)099

References

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[2] C. Vafa, “The String landscape and the swampland,” hep-th/0509212.

G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, “De Sitter Space and the Swampland,” arXiv:1806.08362 [hep-th].

[3] E. Palti, “The Swampland: Introduction and Review,” Fortsch. Phys. 67, no. 6, 1900037 (2019) [arXiv:1903.06239 [hep-th]].

[4] P. Agrawal, G. Obied, P. J. Steinhardt and C. Vafa, “On the Cosmological Implications of the String Swampland,” Phys. Lett. B 784 (2018) 271 [arXiv:1806.09718 [hep-th]].

[5] S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, “De Sitter vacua in string theory,” Phys. Rev. D 68 (2003) 046005 [hep-th/0301240].

[6] J. P. Conlon, F. Quevedo and K. Suruliz, “Large-volume flux compactifications: Moduli spectrum and D3/D7 soft supersymmetry breaking,” JHEP 0508 (2005) 007 [hep-th/0505076].

[7] I. Antoniadis, E. Dudas, S. Ferrara and A. Sagnotti, “The Volkov?Akulov?Starobinsky supergravity,” Phys. Lett. B 733 (2014) 32 [arXiv:1403.3269 [hep-th]].

[8] S. Ferrara, R. Kallosh and A. Linde, “Cosmology with Nilpotent Superfields,” JHEP 1410 (2014) 143 [arXiv:1408.4096 [hep-th]].

[9] I. Antoniadis and C. Bachas, “Branes and the gauge hierarchy,” Phys. Lett. B 450, 83 (1999) [hep-th/9812093].

[10] W. D. Goldberger and M. B. Wise, “Renormalization group flows for brane couplings,” Phys. Rev. D 65 (2002) 025011 [hep-th/0104170].

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[12] K. Becker, M. Becker, M. Haack and J. Louis, “Supersymmetry breaking and alpha-prime corrections to flux induced potentials,” JHEP 0206 (2002) 060 [hep-th/0204254].

[13] S. Gukov, C. Vafa and E. Witten, Nucl. Phys. B 584 (2000), 69-108 doi:10.1016/S0550-3213(00)00373-4 [arXiv:hep-th/9906070 [hep-th]].

[14] I. Antoniadis, S. Ferrara, R. Minasian and K. S. Narain, “R4couplings in M and type II theories on Calabi-Yau spaces,” Nucl. Phys. B 507 (1997) 571 [hep-th/9707013].

[15] E. Kiritsis and B. Pioline, “On R4threshold corrections in IIb string theory and (p, q) string instantons,” Nucl. Phys. B 508, 509 (1997) [hep-th/9707018].

[16] I. Antoniadis, R. Minasian and P. Vanhove, “Noncompact Calabi-Yau manifolds and localized gravity,” Nucl. Phys. B 648 (2003) 69 [hep-th/0209030].

[17] I. Antoniadis, Y. Chen and G. K. Leontaris, “Perturbative moduli stabilisation in type IIB/F-theory framework,” Eur. Phys. J. C 78 (2018) no.9, 766 [arXiv:1803.08941 [hep-th]].

[18] I. Antoniadis, R. Minasian, S. Theisen and P. Vanhove, “String loop corrections to the universal hypermultiplet,” Class. Quant. Grav. 20 (2003) 5079 [hep-th/0307268].

References

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